Is refusing to intentionally break the fast in Ramadan in exchange for obtaining residency, work, and marriage the correct answer, or is breaking the fast for one day to achieve these matters, which help preserve one's faith and perform righteous deeds, better in times of tribulation?

Intentionally breaking the fast in Ramadan (definition) for material gain is not permissible. Whoever intentionally breaks their fast without a valid excuse has committed a grave sin, based on the Prophet Muhammad's (peace be upon him) saying: "Whoever breaks a day of fasting in Ramadan without a concession granted to him by Allah, even if he were to fast for the rest of his life, it would not compensate for it." The pursuit of worldly things should be through obedience to Allah and Taqwa (definition) (piety), for He is indeed the Answerer of prayers.